LAMPLIGHTERS AND THE BOUNDED COHOMOLOGY OF THOMPSON’S GROUP

The paper’s Theorem 3 asserts that for every group G, the lamplighter W = G ≀ Z has vanishing bounded cohomology with separable dual coefficients E in all positive degrees, i.e., H_b^n(W,E)=0 for n>0. The proof proceeds in two stages: (i) the countable case via a Zimmer-amenable Bernoulli action and the Burger–Monod mapping theorem, together with ergodicity with separable coefficients forcing W-equivariant L^∞ cochains to be essentially constant; (ii) a reduction from the general (possibly uncountable) case to the countable case using ℓ^1-homology duality and a Hausdorffness criterion for H_b^1(−,E) (Proposition 12). All of these steps are explicitly documented in Section 2 (in particular, 2.A–2.C and 2.D) of the paper, including the amenable-action setup (Lemma 7 and Corollary 8), the co-amenability transfer (Proposition 9), the mapping theorem computation, and the ergodicity-with-coefficients argument; the countability reduction is handled in 2.D with references to Johnson and Matsumoto–Morita for the ℓ^1-duality/Hausdorff inputs (Theorem 3 statement: ; countable-case argument: ; ergodicity with separable coefficients: ; co-amenability transfer and amenable action stability: ; reduction to countable via ℓ^1-homology/Hausdorffness: ). The candidate solution reproduces exactly this strategy: it builds the same Zimmer-amenable action of W on a Bernoulli space, computes bounded cohomology by W-equivariant L^∞ cochains, uses ergodicity with separable coefficients to force constancy, and then removes countability via ℓ^1-homology duality. The only minor stylistic difference is that the candidate explicitly invokes the Ryll–Nardzewski fixed-point theorem for the Hausdorff step (through Bourbaki), while the paper cites the closed-range/duality route (Johnson; Matsumoto–Morita) to the same end in Proposition 12. There is no mathematical discrepancy. Therefore, both are correct and essentially the same proof at the level of ideas and steps.